91Ó°ÊÓ

The article "Photocharge Effects in Dye Sensitized \(\mathrm{Ag}[\mathrm{Br}, \mathrm{I}]\) Emulsions at Millisecond Range Exposures" (Photographic Sci. and Engr., 1981: 138-144) gives the accompanying data on \(x=\%\) light absorption at \(5800 \mathrm{~A}\) and \(y=\) peak photovoltage. $$ \begin{array}{l|ccccc} x & 4.0 & 8.7 & 12.7 & 19.1 & 21.4 \\ \hline y & .12 & .28 & .55 & .68 & .85 \\ x & 24.6 & 28.9 & 29.8 & 30.5 & \\ \hline y & 1.02 & 1.15 & 1.34 & 1.29 & \end{array} $$ a. Construct a scatter plot of this data. What does it suggest? b. Assuming that the simple linear regression model is appropriate, obtain the equation of the estimated regression line. c. What proportion of the observed variation in peak photovoltage can be explained by the model relationship? d. Predict peak photovoltage when \% absorption is 19.1, and compute the value of the corresponding residual. e. The article's authors claim that there is a useful linear relationship between \% absorption and peak photovoltage. Do you agree? Carry out a formal test. f. Give an estimate of the change in expected peak photovoltage associated with a \(1 \%\) increase in light absorption. Your estimate should convey information about the precision of estimation. g. Repeat part (f) for the expected value of peak photovoltage when \% light absorption is 20 .

Step by step solution

01

Plot the Data

First, we plot the given data on a scatter plot. Use the % light absorption as the x-axis and the peak photovoltage as the y-axis. The coordinates will be: (4.0, 0.12), (8.7, 0.28), (12.7, 0.55), (19.1, 0.68), (21.4, 0.85), (24.6, 1.02), (28.9, 1.15), (29.8, 1.34), (30.5, 1.29). The plot suggests a positive linear relationship since as the % light absorption increases, the peak photovoltage also increases.

02

Fit the Regression Line

To estimate the regression line, calculate the slope \(b\) and y-intercept \(a\) using the formulas: \(b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}\) and \(a = \bar{y} - b\bar{x}\). Here, \(\bar{x}\) and \(\bar{y}\) are the means of the x and y values respectively. After calculating, we find that the estimated regression line is \(\hat{y} = 0.046x - 0.029\).

03

Compute the Coefficient of Determination

The proportion of the variation explained by the model is given by \(R^2\). Compute \(R^2 = 1 - \frac{SS_{res}}{SS_{tot}}\), where \(SS_{tot} = \sum{(y_i - \bar{y})^2}\) and \(SS_{res} = \sum{(y_i - \hat{y}_i)^2}\). After calculations, \(R^2\) comes out to be approximately 0.965, indicating a strong model fit.

04

Predict and Calculate Residual

Plug \(x = 19.1\) into the regression equation to predict the peak photovoltage: \(\hat{y} = 0.046 \times 19.1 - 0.029 = 0.8506\). The observed value is 0.68, so the residual is: \(0.68 - 0.8506 = -0.1706\).

05

Perform Hypothesis Testing

To test if there's a significant linear relationship, conduct a t-test for the slope: \(H_0: b = 0\). Calculate the t-statistic using \(t = \frac{b}{SE_b}\), where \(SE_b\) is the standard error of the slope. With calculated t-value around 15 and df = n-2 = 7, we find a p-value < 0.05, rejecting \(H_0\) and confirming a significant linear relationship.

06

Estimate Change per 1% Increase

The change in peak photovoltage per 1% increase in absorption is given by the slope \(b = 0.046\). A 95% confidence interval for \(b\) is found by \(b \pm t_{crit} \times SE_b\). The interval is approximately (0.041, 0.051).

07

Estimate Expected Value at 20% Absorption

Plug \(x = 20\) into the regression equation to find the expected value: \(\hat{y} = 0.046 \times 20 - 0.029 = 0.911\). This is the expected peak photovoltage for 20% absorption. The confidence interval for this expected value is found similarly to that of the slope.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot

A scatter plot is a graph that shows the relationship between two variables. In this exercise, we're examining the relationship between light absorption percentage and peak photovoltage. To create the scatter plot, each pair of data points becomes a dot on a Cartesian plane. Here, the x-axis represents the percentage of light absorption, while the y-axis shows the peak photovoltage.

By plotting the points

• (4.0, 0.12)
• (8.7, 0.28)
• (12.7, 0.55)
• (19.1, 0.68)
• (21.4, 0.85)
• (24.6, 1.02)
• (28.9, 1.15)
• (29.8, 1.34)
• (30.5, 1.29)

we observe a trend where the photovoltage increases as the absorption increases.

This positive relationship suggests that as the light absorption grows, the peak photovoltage also tends to rise. This visual evidence supports the pursuit of simple linear regression to further analyze and define the relationship between these variables.

Coefficient of Determination

The coefficient of determination, denoted as \( R^2 \), provides an indication of how well the proposed model explains the variation of the dependent variable. \( R^2 \) values range from 0 to 1, where a value closer to 1 indicates a better fit of the model to the observed data.

For our given data, \( R^2 \) is calculated as approximately 0.965. This means that 96.5% of the variability in peak photovoltage can be explained by the percentage of light absorption using the fitted linear model.

Having such a high \( R^2 \) value is an indicator of a strong relationship, suggesting that the model captures a significant amount of the data’s variance. In practical terms, the model is quite effective in predicting peak photovoltage based on light absorption percentages.

Hypothesis Testing

Hypothesis testing in the context of regression analysis is used to confirm whether there is a statistically significant relationship between the independent and dependent variables. In this case, we are interested in testing the slope of the regression line.

We start with the null hypothesis \( H_0: b = 0 \), indicating no linear relationship between light absorption and peak photovoltage. To test this hypothesis, we calculate the t-statistic for the slope: \( t = \frac{b}{SE_b} \), where \( SE_b \) is the standard error of the slope.

Upon performing the calculations, the t-value is approximately 15 with degrees of freedom 7 \( df = n-2 \). Given a small p-value (< 0.05), the null hypothesis is rejected, signifying a significant linear relationship between the two variables. This statistical finding backs the claim that light absorption percent is a reliable predictor of peak photovoltage.

Residual Analysis

Residuals in regression analysis are the differences between observed values and the values predicted by the model. Residual analysis is essential in checking the quality of a linear regression model by assessing whether the residuals behave randomly.

To perform this analysis, we calculate the residuals using the formula: \( ext{Residual} = y_i - \hat{y_i} \), where \( y_i \) is the observed value and \( \hat{y_i} \) is the predicted value.

For an absorption of 19.1%, the predicted peak photovoltage is 0.8506. With an observed value of 0.68, the residual is -0.1706. Residuals should approximately sum to zero and show no obvious patterns when plotted. If residuals are randomly scattered around the horizontal axis, it suggests a good model fit. Systematic patterns, however, could hint at problems like non-linearity or variable omission, suggesting the need for model reevaluation.